Chern Numbers on Positive Vector Bundles and Combinatorics

We study the (k, s)-positivity for holomorphic vector bundles on compact complex manifolds. (0, s)-positivity is exactly the Demailly s-positivity and a (k, 1)-positive line bundle is just a k-positive line bundle in the sense of Sommese. In this way we get a unified theory for all kinds of positivities used for semipositive vector bundles. Several new vanishing theorems for (k, s)-positive vector bundles are proved and the vanishing theorems for k-ample vector bundles on projective algebraic manifolds are generalized to k-positive vector bundles on compact Kähler manifolds.

We prove a classification theorem by cohomology classes for compact Riemannian manifolds with a one-parameter group of isometries without fixed points generalizing the classification of line bundles (more precisely, their circle bundles) over compact manifolds by their first Chern class. We also prove a classification theorem generalizing that of holomorphic line bundles over compact complex manifold by the Picard group of the base for a subfamily of manifolds with additional structure resembling that of circle bundles of such holomorphic line bundles.

A generalization of the notion of almost complex structure is defined on a nonorientable smooth manifold M of even dimension. It is defined by giving an isomorphism J from the tangent bundle TM to the tensor product of the tangent bundle with the orientation bundle such that J○J=−Id TM . The composition J○J is realized as an automorphism of TM using the fact that the orientation bundle is of order two. A notion of integrability of this almost complex structure is defined; also the Kahler condition has been extended. The usual notion of a complex vector bundle is generalized to the nonorientable context. It is a real vector bundle of even rank such that the almost complex structure of a fiber is given up to the sign. Such bundles have generalized Chern classes. These classes take value in the cohomology of the tensor power of the local system defined by the orientation bundle. The notion of a holomorphic vector bundle is extended to the context under consideration. Stable vector bundles and Einstein–Hermitian connections are also generalized. It is shown that a generalized holomorphic vector bundle on a compact nonorientable Kahler manifold admits an Einstein–Hermitian connection if and only if it is polystable.

On any smooth affine surface over an algebraically closed field, it is easy to construct rank two projective modules with given determinant as the first Chern class and an arbitrary rational equivalence class of a zero cycle as the second Chern class. We will briefly indicate this method in Section 1. Here we construct projective modules over smooth affine 3-folds with prescribed Chern classes and of the right rank. That is to say, we construct rank two projective modules with prescribed first and second Chem classes and rank three projective modules with prescribed first, second and third Chern classes (Theorem 2.1). As a corollary, using Roitman's theorem on torsion in zero cycles and Suslin's cancellation theorem, one gets that a rank three projective module over a smooth affine threefold with top Chem class zero has a free direct summand of rank one (Cor. 2.4). For example, over a rational threefold any projective module splits into a free module and a rank two projective module. Thus we obtain necessary and sufficient conditions in terms of appropriate Chern classes for modules to be efficiently generated. For example, on an affine rational 3-fold any line bundle is 3-generated and any rank two projective module is 4-generated. We also prove that any maximal ideal over a smooth affine variety of dimension d is the zero of a section of a rank d projective module (Theorem 3.1). For an affine 3-fold, if A3(X) = 0, we prove the validity of the Eisenbud-Evans estimate for finitely generated modules. We do not know the answers for most of these questions in higher dimensions. Sometimes we assume that the characteristic of the ground field is different from 2, 3 and 5 due to technical reasons.**

Matrices of linear forms of constant rank from vector bundles on projective spaces

We study a version of the Hermitian curvature flow on compact homogeneous complex manifolds. We prove that the solution has a finite extinction time $$T>0$$ and we analyze its behavior when $$t\rightarrow T$$ . We also determine the invariant static metrics and we study the convergence of the normalized flow to one of them.

We present a method for proving the existence of Kiihler-Einstein metrics of positive scalar curvature on certain compact complex manifolds, and use the method to produce a large class of examples of compact Kiihler-Einstein manifolds of positive scalar curvature. Suppose that M is a compact complex manifold of positive first Chern class. As is well-known, the existence of a Kiihler-Einstein metric on M is equivalent to the existence of a solution to a certain complex Monge-Ampere equation on M. To solve this complex MongeAmpere equation by the method of continuity, one needs only to establish the appropriate zeroth order a priori estimate. Suppose now that M does not admit a Kiihler-Einstein metric, so that the zeroth order a priori estimate fails to hold. From this lack of an estimate we extract various global algebro-geometric properties of M by introducing a coherent sheaf of ideals >J on M, called the multiplier sheaf, which carefully measures the extent to which the estimate fails. The sheaf >Y is analogous to the subelliptic multiplier ideal sheaf that J. J. Kohn introduced over a decade ago to obtain sufficient conditions for subellipticity of the d-Neumann problem. Now >J is a global algebro-geometric object on M, and it so happens that >J satisfies a number of highly nontrivial global algebro-geometric conditions, including a cohomology vanishing theorem. In particular, the complex analytic subspace V c M cut out by >J is nonempty, connected, and has arithmetic genus zero. If V is zero-dimensional then it is a single reduced point, while if V is one-dimensional then its support is a tree of smooth rational curves. The logarithmic-geometric genus of M - V always vanishes. These considerations place nontrivial global algebro-geometric restric

In this paper we proved that the Chern numbers of a compact complex manifold covered by a bounded domain have to satisfy certain conditions. For instance, for a compact complex two-fold covered by a bounded domain in the Chern number 2 1 – 2 must be non-negative, where c1 =1st Chern classC2 =2nd Chern class. Some corresponding inequalities for higher dimensional case are also true.

In this paper, by considering a bundle of algebras on a manifold, we construct a vector bundle that naturally contains the tangent bundle of that manifold and has the most important properties of tangent bundles. This vector bundle can be considered as an extended tangent bundle associated to that algebra bundle. To this end, we consider the algebra of the sections of that algebra bundle and find all derivations on this algebra. These derivations are related to the sections of a vector bundle that is the desired bundle. We also investigate the concepts of connection and metric on this extended tangent bundle and prove the existence of the Levi-Civita connection of a metric on the extended tangent bundle.

New techniques are developed, based on the consideration of the projective bundle associated with a direct sum of two vector bundles, to give a simpler solution of the problem of blowing up Chern classes which was previously solved by Porteous[12] using the Grothendieck Riemann-Roch theorem.

This paper focuses on the study of a new category of vector bundles. The objects of this category, called chiral vector bundles, are pairs given by a complex vector bundle along with one of its automorphisms. We provide a classification for the homotopy equivalence classes of these objects based on the construction of a suitable classifying space. The computation of the cohomology of the latter allows us to introduce a proper set of characteristic cohomology classes: some of those just reproduce the ordinary Chern classes but there are also new odd-dimensional classes which take care of the extra topological information introduced by the chiral structure. Chiral vector bundles provide a geometric model for topological quantum systems in class AIII, namely for systems endowed with a (pseudo-)symmetry of chiral type. The classification of the chiral vector bundles over spheres and tori (explicitly computable up to dimension 4), recovers the commonly accepted classification for topological insulators of class AIII which is usually based on the K-group $$K_1$$ . However, our classification turns out to be even richer since it takes care also for possible non-trivial Chern classes.

In this paper, the classical concept of Weierstrass points on a Riemann surface is generalized to the consideration of similar points associated with a holomorphic vector bundle E over a compact complex manifold M. These points are invariants of the pair (E, M). The study of these general Weierstrass points is then initiated in this paper by deriving some results about the relationship of the points to singular sets of holomorphic mappings of the manifold to Grassmann spaces associated with the vector space of sections of the vector bundle. The accessibility of the point sets are demonstrated with some examples.

The Griffiths conjecture asserts that every ample vector bundle $E$ over a compact complex manifold $S$ admits a hermitian metric with positive curvature in the sense of Griffiths. In this article we give a sufficient condition for a positive hermitian metric on $\mathcal{O}_{\mathbb{P}(E^*)}(1)$ to induce a Griffiths positive $L^2$-metric on the vector bundle $E$. This result suggests to study the relative K\ahler-Ricci flow on $\mathcal{O}_{\mathbb{P}(E^*)}(1)$ for the fibration $\mathbb{P}(E^*)\to S$. We define a flow and give arguments for the convergence.

In this paper, the classical concept of Weierstrass points on a Riemann surface is generalized to the consideration of similar points associated with a holomorphic vector bundle E over a compact complex manifold M. These points are invariants of the pair (E, M). The study of these general Weierstrass points is then initiated in this paper by deriving some results about the relationship of the points to singular sets of holomorphic mappings of the manifold to Grassmann spaces associated with the vector space of sections of the vector bundle. The accessibility of the point sets are demonstrated with some examples.

IfX is a compact complex manifold of complex dimension two, denote by K x the canonical line bundle (the second exterior power of the cotangent bundle) of X. It is immediate from the classification of surfaces [15, 5] that if K x is negative then X is either the complex projective plane IP 2, or the product tP ~ x tP 1 of two projective lines ( t h e non-singular quadric surface Q2C IP3), or is derived from one of these by blowing up points. Here a vector bundle E on a complex space X (singularities allowed) is negative if the zero section X o is exceptional in E. Now if (X, (gx) is an analytic surface with singular points, then if each singularity is Cohen-Macaulay (the homological codimension of each stalk (gx, x equals the dimension of X = 2; this is equivalent to normal (integrally closed) in this dimension) then at least the canonical sheaf ~ x is defined and Serre-Grothendieck duality holds. And if each point is Gorenstein ((gx, x has finite injective dimension) then J{x is locally trivial and so it is the sheaf of sections of a holomorphic line bundle Kx, the canonical bundle of the singular space X. The purpose of this note is to expose all compact Gorenstein surfaces (compact two-dimensional complex spaces with only Gorenstein singularities) for which K x is negative.